This paper is devoted to the Čech covering dimension of the Sorgenfrey plane \(S\times S\) . (We use the
standard notation \(S\)  for the Sorgenfrey line and assume that the underlying set of \(S\)  is \(\mathbb {R}\) .) Obviously,
\(\operatorname{ind}S=\operatorname{ind}S^\omega =0\) , \(\operatorname{Ind}S\times S = \infty \)  (because \(S\times S\)  is not normal), and \(\dim S\times S> 0\) 
(for the same reason). It is also known that \(\dim S=0\) , because \(S\)  is Lindelöf and \(\dim \leqslant \operatorname{ind}\)  for
Lindelöf spaces (see, e.g., [1]}), and that \(\dim _0 S^\omega =0\)  and hence
\(\dim _0 S^n =0\)  for all \(n\in \omega \)  [2]}. Moreover, \(\dim _0 S^\kappa =0\)  for any cardinal \(\kappa \) ;
Terasawa credited this result to K. Morita in [2]}, and its proof can be found in [4]}. We prove
that \(\dim S\times S=\infty \) .
